Problem Solving Techniques in Mathematics: Handshakes, Sequences, and Classroom Seating

1. 5-Minute Check on Activity 1-1 • Problem Solving Techniques: • What do we need to get from the information given to us about a problem? • What was the problem from yesterday? • What did we do to help solve the problem? All relevant information on the problem Do we have time to get the new book before class starts? We drew a graph (model) with a time-line. Click the mouse button or press the Space Bar to display the answers.

2. Activity 1 - 2 The Handshake

3. Objectives • Organize information • Develop problem-solving strategies • Draw a picture • Recognize a pattern • Do a simpler problem • Communicate problem-solving ides

4. Vocabulary • Arithmetic sequence– a list of numbers in which consecutive numbers share a common difference • Geometric sequence– a list of numbers in which consecutive numbers share a common ratio • Fibonacci sequence– a list of numbers in which consecutive numbers are added to get the next number • Inductive reasoning– arrives at a general conclusion form specific examples • Deductive reasoning– uses laws and properties to prove/disprove conjectures

5. 4 Steps in Problem Solving • Understand the problem (determine what’s involved) • Devise a plan (look for connections to obtain the idea of a solution) • Carry out the plan • Look back at the completed solution (review and discuss it) from George Polya’s book, How to Solve It

6. Problem Solving Summary • Problem Solving Strategies • Discussing the problem • Organizing information • Drawing a picture • Recognizing patterns • Doing a simpler problem • Four-step process • Understand the problem • Devise a plan • Carry out the plan • Look back at the completed solution

7. Description This mathematics course involves working with other students in the class, so form a group of 3, 4, or 5 students. Introduce yourself to every other student in your group with a firm handshake. Share some information about yourself with the other members of your group.

8. Handshakes Table 1 3 6 10 15 21 2 students: A  B 3 students: A  B, A  C, and B  C  4 students: A  B, A  C, A  D, B  C, B  D, and C  D    

9. Handshakes Describe a rule for determining the number of handshakes in a group of A. seven studentsB. students in our classC. n students Start adding 1 less than 7 and 2 less than 7 until you get down to 1: 6 + 5 + 4 + 3 + 2 + 1 = 21 handshakes Start adding 1 less than x and 2 less than x until you get down to 1: Start adding 1 less than n and 2 less than n until you get down to 1:

10. The Classroom • Square tables in a classroom could be arranged in various clusters • Figure out the number of students at each cluster

11. Classroom Table 4 6 8 10 12 14 16 A 1 table: A, B, C, D 1 B D A E C 1 2 2 tables: A, B, C, D and E, F B D C F A E H 1 2 3 3 tables: A, B, C, D, E, F and H, G B D C F G

12. Class Room Seating Describe the pattern that connects the number of square tables in a cluster and the total number of students that can be seated. Write a rule in sentences that will determine the total number of students that can site in a cluster of a given number of square tables, n. Pattern: add a table, add two students Rule: Starting with 4 students at one table, with every table you add, you add two more students. Math: 4 + 2(n – 1)

13. Class Room Seating 24 students are in a science course at MSHSa. How many tables must be put together to seat a group of 6 students?b. How many clusters of tables are needed for that class? How would we use square table clusters in our class? 2 24  6 = 4

14. Summary and Homework • Summary • Problem Solving Strategies • Discussing the problem • Organizing information • Drawing a picture • Recognizing patterns • Doing a simpler problem • Four-step process • Understand the problem • Devise a plan • Carry out the plan • Look back at the completed solution • Homework • pg 6-9; 1-7, 9